Logical Aspects of Artificial Intelligence Introduction to Description Logics

Stephane´ Demri [email protected]

December 9th, 2019 What is in this part of the course?

Introduction to Description Logics and Temporal Logics for Multi-Agent Aystems

I Today: Introduction to description logics.

I 16/12/2019: Tableaux calculi and complexity.

I 06/01/2020: Introduction to temporal logics for multi-agent systems.

I 13/01/2020: 14h00–16h00 Exam on this part of the course (slides allowed). but also first-order logic, modal logics, knowledge representation, etc... 2 What can you expect to learn?

I Basics of description logic including ALC as well as ATL and variants.

I Tableaux for ALC, model-checking techniques for ATL-like logics.

I Complexity, decidability, expressive power results for logics dedicated to knowledge representation.

3 Background

1. Necessary background

I Basics of first-order logic.

I Basics of complexity theory.

2. Optional background

I Basics of modal logics, temporal logics

I Sequent-style proof systems.

I Basics of model-checking.

4 Course material

I Slides and exercises available on https://wikimpri.dptinfo.ens-cachan.fr/doku. php?id=cours:c-1-39 http://www.lsv.fr/˜demri/notes-de-cours.html

I Slides available after each lecture.

I Do not hesitate to contact me ([email protected]).

5 Material mainly based on the following documents

I F. Baader, I. Horrocks, C. Lutz and U. Sattler. An introduction to Description Logic. Cambridge University Press, 2017.

I Ivan Varzinczak’s slides (ESSLLI’18)

I Ulrike Sattler & Thomas Schneider’s slides (ESSLLI’15).

I S. Demri, V. Goranko, M. Lange. Temporal Logics in Computer Science. Cambridge University Press, 2016.

6 Other (online) ressources

I Description Logic Complexity Navigator by Evgeny Zolin. http://www.cs.man.ac.uk/˜ezolin/dl/

I Proceedings of the Description Logic Workshops http://dl.kr.org/workshops/

I See also the proceedings of the international conferences:

I Int. Joint Conference on Artificial Intelligence. (IJCAI) I European Conference on Artificial Intelligence. (ECAI) I Int. Conference on Principles of Knowledge Representation and Reasoning. (KR)

7 Plan of the lecture

I Knowledge representation.

I Basic description logic ALC.

I Several extensions of ALC.

I Relationships with first-order logic and modal logics.

I Exercises session.

8 Knowledge representation

9 DLs: where they come from

I First-order logic is not always the most natural language.

∀x ∃ y ∀z ((P(x, y) ∧ Q(y, z) ⇒ (¬Q(a, y) ∨ P(x, z))))

∀ x (Teacher(x) ⇔ Person(x)∧∃ y (Teaches(x, y)∧Course(y)))

I How to design user-friendly languages for knowledge representation ?

I Concept definition from Description Logics.

Teacher ≡ Person u ∃Teaches.Course

10 Reasoning about . . .

I Knowledge Epistemic Logics

I Rules and obligations Deontic Logics

I Programs Hoare Logics

I Time Temporal Logics but also many-valued logics, non-monotonic logics, team logics, separation logics, etc.

11 Ontologies

I Formal specification of some domain with concepts, objects, relationships between concepts, objects, etc.

I Backbone of ontologies includes: I taxonomy (classification of objects), I axioms (to constrain the models of the defined terms).

I Classification of medical terms: diseases, body parts, drugs, etc.

I Well-known ontologies: I Medical ontology SNOMED-CT formalised with description logic EL + +. I NCI Thesaurus (National Cancer Institute, USA). I Gene ontology (world largest source of information on the functions of genes).

I Free ontology editor Proteg´ e´ http://protege.stanford.edu/

12 The classical student ontology

I Natural language specification:

I Employed students are students and employees.

I Students are not taxpayers.

I Employed students are taxpayers.

I Employed students who are parents are not taxpayers.

I To work for is to be employed by.

I John is an employed student, John works for IBM.

I Classes/relations/individuals.

13 Main ingredients in formal ontologies

I Model of the world with classes within a domain, relationships between classes and instantiations of classes.

I Formal: abstract model of some domain with (mathematical) semantics and reasoning tasks.

I Classes or concepts: classes of objects with the domain of interest. (Employed student, Parent, Course)

I Relations or roles: relationships between concepts. (being employed by, sibling-of)

I Instances of classes and relations.

I John is an employed student. I Mary works for IBM.

14 Early KR formalisms

I Graphical formalisms easier to grasp and supposedly close to how knowledge is represented by human beings.

I Large variety of semantical networks.

I Often, lack of formal semantics (see tentatives with the knowledge representation system KL-ONE).

15 Why Description Logics?

I Formal languages for concepts, relations and instances.

I DLs have all one needs to formalise ontologies.

I Computational properties.

I Acceptable trade-off between expressivity and complexity. I Decidability and often tractability. I Implementation in tools of the main reasoning tasks.

I A remarkable suite of languages and tools. See e.g.

I OWL: Web Ontology Language. I Proteg´ e:´ ontology editor. I FaCT++: DL reasoner supporting OWL DL.

16 Description Logics and Knowledge representation

I Description is a subfield of Knowledge Representation, itself a subfield of Artificial Intelligence.

I Description Logic(s):

I a research field, I a family of knowledge representation languages, I a member of the family.

I Well-defined syntax with formal semantics, decision problems, algorithms, etc.

17 Basic description logic ALC

18 DLs: the core

I Concept language.

Person u ∃Teaches.Course

I Syntactic ingredients of the concept language:

I Concept names for sets of elements, e.g. Person. I Role names interpreted by binary relations between objects, e.g. EmployedBy. I Concept constructors to build complex concepts, e.g. ¬, u, t, ∃.

I Basic terminology stored in a TBox.

I Facts about individuals stored in an ABox.

19 Basic elements of the language

I Concept names.

def NC = {A1, A2,...}

Examples: Parent, Sister, Student

I Role names. def NR = {r1, r2,...} Examples: EmployedBy, MotherOf

I Individual names.

def NI = {a1, a2,...}

Examples: Mary, Alice, John

20 Boolean constructors & role restrictions

I Boolean constructors.

I Concept negation ¬ (class complement)

I Concept conjunction u (class intersection)

I Concept disjunction t (class union)

I Role restrictions. I Existential restriction ∃ (at least one related individual)

I Value restriction ∀ (all related individuals)

I Many more constructors exist, see forthcoming ALC extensions.

I For modal logicians, ¬, u, t, ∃, ∀ ∼ ¬, ∧, ∨, 3, 2

21 Complex concepts in ALC

I ALC: Attributive Concept Language with Complements.

I Complex concepts. C ::= > | ⊥ | A | ¬C | C u C | C t C | ∃r.C | ∀r.C,

where A ∈ NC and r ∈ NR.

I Examples of complex concepts:

I Student u ¬∃Pays.Tax I ∃MotherOf.(∃MotherOf.A)

I Syntax errors in Student t ∀¬Tax ∀∃MotherOf.Mary

def I C ⇒ D = ¬C t D.

22 Interpretation

I Concept/role/individual ∼ unary predicate/binary predicate/constant.

def I I I Interpretation I = (∆ , · )

I I ∆ : non-empty set (the domain).

I I · : interpretation function such that

AI ⊆ ∆I r I ⊆ ∆I × ∆I aI ∈ ∆I

I I A priori, ∆ is arbitrary and I can be viewed as a first-order model for unary and binary predicate symbols and constants.

23 Semantics for complex concepts

>I =def ∆I ⊥I =def ∅ (¬C)I =def ∆I \ CI I def I I (C1 t C2) = C1 ∪ C2 I def I I (C1 u C2) = C1 ∩ C2 (∃r.C)I =def {a ∈ ∆I | r I (a) ∩ CI 6= ∅} (∀r.C)I =def {a ∈ ∆I | r I (a) ⊆ CI }

(R(a) =def {b | (a, b) ∈ R})

I I In modal logic lingua, a ∈ C corresponds to I, a |= C.

24 Graphical representation

Teaches

C2 b Course Course Person Teaches

Teaches a C1 Person Course Teacher

I I ∆ = {a, b, C1, C2}. I I Teaches = {(a, C1), (a, C2), (b, b)}. I I I Person = {a, b}, Course = {C1, C2, b}. I I a ∈ (∀Teaches.Course) . 25 Concept satisfiability problem

I Concept satisfiability problem: Input: A (complex) concept C in ALC. Question: Is there an interpretation I = (∆I , ·I ) such that CI 6= ∅?

I This corresponds to the standard formulation for the satisfiability problem (in modal logics, temporal logics, etc.).

I The concept satisfiability problem for ALC is PSPACE-complete.

I ALC has the finite interpretation property: every satisfiable concept has an interpretation with a finite domain.

26 Statements

I Concept inclusion. Teachers are persons. Employed students are employees.

I Concept and role membership. Mary is a student. Alice is a teacher. Laura teaches the course “Automata Theory”.

I Statements are not concepts and express properties of concepts, roles and individuals.

27 General concept inclusion (GCI)

I Expressions of the form

C v D

are called general concept inclusion.

I Intuitive meaning:

I D subsumes C. I C is more specific than D.

I Example: Employee v ∃WorksFor.>.

def I I I Satisfaction relation: I |= C v D ⇔ C ⊆ D .

I C v D understood as a global statement about I.

28 Concept equivalence

I C v D and D v C abbreviated by

C ≡ D

called concept equivalence.

def I I I Satisfaction relation: I |= C ≡ D ⇔ C = D .

I Concept definition (A ∈ NC is a concept name) A ≡ C

I > ≡ (¬Student t Student).

29 Subsumption problem

I Subsumption problem: Input: A GCI C v D with C, D ∈ ALC. Question: Is it the case that for all interpretations I, we have I |= C v D?

I C v D is “not valid” iff C u ¬D is satisfiable.

I As coPSPACE =PSPACE, the subsumption problem for ALC is PSPACE-complete too.

30 Assertions

I Concept assertion: stating that an individual is an instance of a concept. a : C

def I I I Satisfaction relation: I |= a : C ⇔ a ∈ C .

I Role assertion: stating that two individuals are in a relation. (a, b): r def I I I I Satisfaction relation: I |= (a, b): r ⇔ (a , b ) ∈ r .

I Examples: I Alice : Student u ¬∃Pays.Tax. I (Laura, CNRS): WorksFor.

31 The validity problem

I Validity problem: Input: A statement α in ALC. Question: Is the case that for all interpretations I, we have I |= α?

I Validity of α is written |= α.

I Validity of > v C corresponds to the usual notion of validity for C, i.e. for all interpretations I = (∆I , ·I ), we have CI = ∆I .

I Examples of valid statements:

I |= ∀r.C u D v ∀r.C. I |= a : C t ¬C. I |= > v (¬(C u D) t (C t D)).

I The validity problem for ALC is PSPACE-complete.

32 What is a knowledge base (a.k.a. ontology) ?

I Terminological Box (TBox) T : finite collection of GCIs.

I I.e., a finite set of subsumption statements. I This provides definitions of concepts (a terminology).

I Assertional Box (ABox) A: finite collection of assertions.

I I.e., a finite set of concept and role assertions. I This provides a partial view on the interpretations and can be understood as a finite database.

I Knowledge base K is a pair (T , A).

I Knowledge bases are also called ontologies.

33 A knowledge base K?

I TBox T : Course v ¬Person Teacher v Person u ∃Teaches.Course ∃Teaches.> v Person Student v Person u ∃Attends.Course ∃Attends.> v Person

I ABox A: Mary : Person CS600 : Course Alice : Person u Teacher (Alice, CS600): Teaches (Mary, CS600): Attends

34 Consequences from knowledge bases

I I I Interpretation I = (∆ , · ), knowledge base K = (T , A). def I I |= A ⇔ for all α ∈ A, we have I |= α. def I I |= T ⇔ for all α ∈ T , we have I |= α. def I I |= K ⇔ I |= A and I |= T .

def I K |= α ⇔ for all interpretations I such that I |= K, we have I |= α.

I K? |= CS600 : ¬Person and K? |= Alice : Teacher.

35 Decision problems relatively to a knowlegde base

def I K is consistent ⇔ there is some I such that I |= K.

def I C is satisfiable with respect to K ⇔ there is I such that I |= K and CI 6= ∅.

def I C is subsumed by D with respect to K ⇔ K |= C v D.

def I C and D are equivalent with respect to K ⇔ T |= C ≡ D.

def I a is an instance of C with respect to K ⇔ K |= a : C.

I K |= C v D also written C vK D. K |= C ≡ D also written C ≡K D.

36 Relationships between reasoning problems

I C and D are equivalent w.r.t. K iff C is subsumed by D w.r.t. K and D is subsumed by C w.r.t. K.

I C vK D iff C u ¬D is not satisfiable w.r.t. K.

I C is satisfiable w.r.t. K iff C 6vK⊥.

I C is satisfiable w.r.t. K iff (T , A ∪ {b : C}) is consistent. (b is fresh)

I K |= a : C iff (T , A ∪ {a : ¬C}) is not consistent.

37 C is satisfiable w.r.t. K iff (T , A ∪ {b : C}) is consistent

I Suppose that C is satisfiable w.r.t. K. I I I There is I such that I |= K and C 6= ∅, say a ∈ C .

0 I0 def I Let I be the variant of I such that b = a.

0 I As b does not appear in K and C, we have I |= K.

0 I I0 I Furthermore, I |= b : C as C = C .

0 I Consequently, I |= (T , A ∪ {b : C}).

I Now, suppose that (T , A ∪ {b : C}) is consistent.

I There is I such that I |= T , I |= A and I |= b : C.

I I I Consequently, b ∈ C .

I I So, there is some I such that I |= K and C is non-empty.

38 Classification

I Deduce implicit knowledge from the explicitly represented knowledge.

I For all A, B in K, check whether A vK B.

I For all A in K, check whether A is satisfiable w.r.t. K. If not for some B, a modelling error is probable.

I For all a and C in K, check whether K |= a : C.

I Classifying a knowledge base K. 1. Check whether K is consistent, if yes, go 2. 2. For each pair A, B of concept names (plus >, ⊥), check whether K |= A v B. 3. For individual name a and concept C in K, check whether K |= a : C. leading to K’s inferred class hierarchy. 39 Complexity results for ALC

I Concept satisfiability and subsumption problems are PSPACE-complete. (no knowledge base involved)

I Knowlegde base consistency problem is EXPTIME-complete.

NP ⊆ PSPACE ⊆ EXPTIME ⊂ 2EXPTIME ⊂ N2EXPTIME

I Recall that C vK D iff (T , A ∪ {b : C u ¬D}) is not consistent.

40 Several extensions of ALC

41 Extensions: a feature of DLs

I Concepts/assertions in ALC have a limited expressive power.

I How to express simple arithmetical constraints such as “Alice teaches at least three courses”? I How to enforce constraints between roles? For instance, r I = (sI )−1 or r I ⊆ sI .

I The expressive power of ALC concepts can be characterised precisely, thanks to the notion of bisimulation (not presented today).

B Trade-off between the expressive power and the computational properties of the extensions.

I In the other direction: study of ALC fragments to reduce the complexity while preserving the expression of interesting properties, see e.g. EL or FL0.

42 Inverse roles Course v ¬Person Teacher v Person u ∃Teaches.Course ∃Teaches.> v Person Student v Person u ∃Attends.Course ∃Attends.> v Person Professor v Teacher Course v ∀TaughtBy.¬Professor

I Extending NR with inverse roles: − NR ∪ {r | r ∈ NR}

def I I − I def I −1 I Given I = (∆ , · ), (r ) = (r ) where R−1 =def {(b, a) | (a, b) ∈ R}

43 Elimination of the role name TaughtBy

I Back to the previous example. Professor v Teacher Course v ∀Teaches−.¬Professor

I Given a logic L, LI is defined as L except that inverse roles are added.

I Concept satisfiability for ALCI remains PSPACE-complete and knowledge consistency remains EXPTIME-complete.

44 Number restrictions

I How to express in ALC that a student attends to at least three courses? Student v ∃Attends.Course u A ∃Attends.Course u ¬A u B ∃Attends.Course u ¬A u ¬B

I Why isn’t it satisfactory ?

I How to express in ALC that a student attends to at most 10 courses?

I There is no concept C in ALC such that for all interpretations I, for all a ∈ ∆I , a ∈ CI iff card({b | (a, b) ∈ AttendsI }) ≥ 3

45 (Unqualified) number restriction

I Extending the concepts with number restrictions (≤ n r) and (≥ m r).

def I I I Given I = (∆ , · ),

(≤ n r)I =def {a ∈ ∆I | card({b | (a, b) ∈ r I }) ≤ n}

(≥ m r)I =def {a ∈ ∆I | card({b | (a, b) ∈ r I }) ≥ m}

I Given a logic L, LN is defined as L except that (unqualified) number restrictions are added.

I In ALCN , (≥ 3 Attends) u (≤ 10 Attends) does the job.

46 Qualified number restriction

I Generalising the number restrictions.

I Qualified number restrictions: (≤ n r · C), (≥ m r · C).

def I I I Given I = (∆ , · ),

def (≤ n r·C)I = {a ∈ ∆I | card({b | (a, b) ∈ r I and b ∈ CI }) ≤ n}

def (≥ m r·C)I = {a ∈ ∆I | card({b | (a, b) ∈ r I and b ∈ CI }) ≥ m}

I (∼ n r) = (∼ n r · >).

I Given a logic L, LQ is defined as L except that qualified number restrictions are added.

I Concept satisfiability for ALCIQ is PSPACE-complete and knowledge base consistency is EXPTIME-complete.

47 Naming individuals in complex concepts

I Concepts in ALC C ::= > | ⊥ | A | ¬C | C u C | C t C | ∃r.C | ∀r.C

I . . . but individual names a (as in the concept assertion a : C) are not concepts in ALC.

I How to express the concept “the courses taught by Alice” ? Course u ∃Teaches−.Mary

I Nominals in hybrid (modal) logics: propositional variables true in only one world of the domain.

I Nominals in DLs: individual names inside concept I def I descriptions, written {a}, where a ∈ NI with {a} = {a }. B Syntactic trick {·}: Course u ∃Teaches−.{Mary}.

48 Nominals in DLs

I Given a logic L, LO is defined as L except that nominals are added.

I Concept satisfiability for ALCOQ is PSPACE-complete and knowledge base consistency is EXPTIME-complete.

I . . . but concept satisfiability for ALCOI is EXPTIME-complete.

49 Role hierarchies

I ALC is not able to express complex role constraints such that a relation is included in another relation.

I Typically, the interpretation of Attends should include the interpretation of AttendsActively.

I Role inclusion axiom (RIA) of the form r v s with

I |= r v s ⇔def r I ⊆ sI

I Given a logic L, LH is defined as L except that role inclusion axioms are added (in the TBox).

I Concept satisfiability for ALCH is PSPACE-complete and knowledge base consistency is EXPTIME-complete.

50 Role value maps are concepts!

I A role value map is an atomic concept of the form r v s:

(r v s)I =def {a ∈ ∆I | r I (a) ⊆ sI (a)}

B Role value maps are local variants of role inclusion axioms (RIAs).

I The RIA r v s can be encoded by the GCI > v (r v s).

51 Transitive roles

I Many natural relations are transitive (AncestorOf, HasPart, etc.) but this cannot be expressed in ALCH.

I Transitivity axioms are of the form Trans(r):

I |= Trans(r) ⇔def r I ◦ r I ⊆ r I

I Given L, its extension with transitivity axioms in TBoxes is obtained by replacing ALC by S (new naming rule).

I Concept satisfiability for S is PSPACE-complete and EXPTIME-complete for knowledge base consistency.

I Other properties are included in knowledge bases such as reflexivity, irreflexivity, symmetry, functionality, . . .

52 A selection of complexity results

53 A queen logic SROIQ

I SROIQ agrees with the ontology language OWL 2 DL.

I SROIQ contains more than one might think from its name.

I Its knowledge bases contain a role box (RBox) to specify constraints about the interpretation of the role expressions.

I The set of roles R is made of role names r, its converses r − and the universal role U with UI =def ∆I × ∆I .

I New atomic concept ∃R.Self with (∃R.Self)I =def {a | (a, a) ∈ RI }

I Role assertions (a, b): ¬R are allowed in the ABox.

I + ingredients the name (nominals, qualified number restrictions, inverse). 54 Role axioms in the RBox

I Complex role inclusion axioms (CRIA) R1 ◦ · · · ◦ Rn v R.

B A regularity constraint is required on the set of CRIAs (unspecified here).

I Role axioms specifying disjointness, transitivity, reflexivity, irreflexivity, symmetry, asymmetry.

I The knowledge base consistency problem for SROIQ is N2EXPTIME-complete.

55 Semantic Web

I Semantic web:

I A vision of a computer-understandable Web. I Distributed knowledge and data in reusable form. I XML, RDF and OWL are part of the story.

I Principles towards a semantic Web of data

I Give a name to everything. I Relationships form a graph between the entities. I The names are addresses on the Web. I Provide a formal semantics so that knowledge is encoded in a machine interpretable way.

56 OWL based on description logics

I OWL: Web Ontology Language.

I Motivated by semantic web activities: add meaning to web content by annotating it with terms defined in ontologies.

I It is a World Wide Web (W3C) standard.

I OWL has an explicit formal semantics.

I Supported by tools and infrastructures such as development environments, reasoners and information systems.

I Based on description logics such as SROIQ.

57 DLs and ontology languages

I W3C’s OWL 2 is based on SROIQ.

I OWL 2 EL based on EL and OWL 2 QL based on DL-Lite.

I OWL was based on SHOIN .

I An OWL ontology is a mixed set containing TBox axioms and ABox assertions.

I More on complexity/scalability:

I OWL (SHOIN ) is NEXPTIME-complete. I OWL 2 EL is PTIME-complete.

58 OWL RDF/XML exchange syntax

Person u ∀Teaches.Course

59 OWL reasoners and Proteg´ e´

I OWL reasoners: implement decision procedures for consistency and ontology classification.

I Open-source ontology editor Proteg´ e.´

I Interaction with DL reasoners (FaCT++, Pellet, Racer) via the OWL API.

I Show results about ontology classification.

I Helpful to work with toy ontologies.

60 Relationships with other logics

61 Foreign language for modal/classical logicians

I Description logics can be understood as fragments of first-order logics.

I Similarly, reasoning tasks for description logics can be understood as decision problems for modal logics.

I This can be made precise and sometimes results for modal/first-oder logics can be used.

I Specificity of DLs: many fragments, many extensions and numerous original reasoning tasks (non-exhaustive presentation in this course).

62 Description logics and its shared history

I Late 80s: description logics developped as logical formalisms for semantics networks.

I In the 1990s: relationships with first-order logic, modal logics, PDL-like logics, etc. (PDL = Propositional Dynamic Logic)

I Logical basis for the Web Ontology Language OWL.

I Analogies between ontologies and databases lead also to relationships with query answering languages.

63 From concepts to first-order formulae

I I I Interpretations I = (∆ , · ) understood as first-order models.

I Translation of non-logical symbols.

Description logics First-order logic individual name a ∈ NI ≈ constant a concept name A ∈ NC ≈ unary predicate A role name r ∈ NR ≈ binary predicate r

I Translation of concepts, assertions and GCIs by internalising the DL semantics.

64 Example of translation

I A small TBox and its small ABox: ∃Attends.> v Person Teacher ≡ Person u ∃Teaches.Course Alice : Teacher

I Its translation in FOL:

∀ x (∃ y Attends(x, y) ⇒ Person(x)) ∧ ∀ x (Teacher(x) ⇔ (Person(x)∧∃ y(Teaches(x, y)∧Course(y)))) ∧ Teacher(Alice)

I Concepts can be understood as first-order formulae with one free variable.

65 Internalisation of ALC semantics

t(A, x) =def A(x) t(>, x) / t(⊥, x) =def > / ⊥ t(¬C, x) =def ¬t(C, x) def t(C1 u C2, x) = t(C1, x) ∧ t(C2, x) def t(C1 t C2, x) = t(C1, x) ∨ t(C2, x) t(∃r.C, x) =def ∃ y r(x, y) ∧ t(C, y) t(∀r.C, x) =def ∀ y r(x, y) ⇒ t(C, y) where y is a fresh variable. I I I I Given I = (∆ , · ), a ∈ C iff I, ρ[x ← a] |= t(C, x).

I Many-one reduction from ALC satisfiability problem to FOL satisfiability.

66 Translating KBs

t(C v D) =def ∀ x t(C, x) ⇒ t(D, x) t(C ≡ D) =def ∀ x t(C, x) ⇔ t(D, x) t(a : C) =def t(C, x)[a/x] t((a, b): r) =def r(a, b) where ϕ[a/x] (also written ϕ(a)) is the formula obtained from ϕ obtained by replacing the free occurrences of x by a.

I t(K) is defined as the conjunction below: ^ ^ t(α) ∧ t(α) α∈T α∈A

I I I Given I = (∆ , · ), I |= K iff I |= t(K).

67 Locating DLs within first-order logic fragments

I The definition of t(C, x) can be optimised to recycle variables and to use only two variables x0 and x1. def t(∃r.C, xi ) = ∃ x1−i r(xi , x1−i ) ∧ t(C, x1−i ) def t(∀r.C, xi ) = ∀ x1−i r(xi , x1−i ) ⇒ t(C, x1−i )

I FO2 (FOL restricted to two individual variables) satisfiability is NEXPTIME-complete.

I Actually, recycling of variables leads to the guarded fragment restricted to two variables GF2, whose satisfiability is EXPTIME-complete.

I Which additional DL features can be translated into FOL? into a decidable fragment of FOL?

68 More translations into FOL

t(∃r −.C, x) =def ∃ y r(y, x) ∧ t(C, y) t({a}, x) =def a = x t((≥ n r · C), x) =def ∃≥n y r(x, y) ∧ t(C, y) t(∃r.Self, x) =def r(x, x) t(r v s) =def ∀ x, y r(x, y) ⇒ s(x, y) def t(Trans(r)) = ∀x1, x2, x3 r(x1, x2) ∧ r(x2, x3) ⇒ r(x1, x3) def t(r1 ◦ · · · ◦ rn v r) = ∀x1, ··· , xn+1 r1(x1, x2) ∧ · · · ∧ rn(xn, xn+1) ⇒ r(x1, xn+1)

≥k def ^ ^ ∃ x ϕ(x) = ∃ x1,..., xk ( xi 6= xj ) ∧ ϕ(xi ) i6=j i

69 Modal logics in a nutshell

I Formulae: ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | 3ϕ | 2ϕ.

I Kripke-style structures M = (W , R, V ):

I W : non-empty set of worlds. I R ⊆ W × W : accessibility relation. I V : PROP → P(W ): valuation. p |= 33p ∧ 33¬p ∧ 2¬p w q p, q q

I Satisfaction relation: def I M, w |= p ⇔ w ∈ V (p). def 0 0 0 I M, w |= 3ϕ ⇔ there is w s.t. (w, w ) ∈ R and M, w |= ϕ. def 0 0 0 I M, w |= 2ϕ ⇔ for all w s.t. (w, w ) ∈ R, M, w |= ϕ.

70 Ubiquity of modal logics

I Satisfiability problem: given a formula ϕ, are there M, w such that M, w |= ϕ?

I Plethora of modal logics depending on the frame conditions:

I Modal logic S5: R is an equivalence relation (or R = W × W ). I Modal logic K: R is arbitrary (or (W , R) is a finite tree). I Modal logic S4: R is reflexive and transitive.

I Epistemic/temporal logics can be viewed as modal logics with

I specific frame conditions (e.g., (W , R) is a tree), I multiple modalities (e.g., [r] for r ∈ NR), I modalities of arity > 1 (e.g., the until operator U). ϕUψ, ϕ ϕ ϕ ψ ...

71 Adding the universality modality and nominals

ϕ ::= A | a | ¬ϕ | ϕ ∧ ϕ | hriϕ | [r]ϕ | [U]ϕ | hUiϕ

I Models of the form M = (W , (Rr )r∈NR , V ) with associated modalities hri and [r].

def I University modality [U] such that M, w |= [U]ϕ ⇔ for all w 0, we have M, w 0 |= ϕ.

I Propositional variables are now denoted by A, B to prepare the encoding.

I Nominals in hybrid modal logics are propositional variables holding for a unique world, here represented by a, b.

I Satisfiability problem for this logic is EXPTIME-complete.

72 Internalisation of the semantics t(A) =def A t(>) / t(⊥) =def > / ⊥ t(¬C) =def ¬t(C) def t(C1 u C2) = t(C1) ∧ t(C2) def t(C1 t C2) = t(C1) ∨ t(C2) t(∃r.C) =def hrit(C) t(∀r.C) =def [r]t(C)

I I I I = (∆ , · ) 7→ MI = (W , (Rr )r∈NR , V ) def I I I W = ∆ and Rr = r for all r. def I def I I V (A) = A for all A and V (a) = {a } for all a.

I I I I Given I = (∆ , · ), a ∈ C iff MI , a |= t(C).

I Many-one reduction from ALC satisfiability problem to modal satisfiability.

73 Translating KBs

t(C v D) =def [U](t(C) ⇒ t(D)) t(C ≡ D) =def [U](t(C) ⇔ t(D)) t(a : C) =def hUi(a ∧ t(C)) t((a, b): r) =def hUi(a ∧ hrib)

I t(K) is defined as the conjunction below: ^ ^ t(α) ∧ t(α) α∈T α∈A

I I I I Given I = (∆ , · ), I |= K iff MI , a |= t(K) for all a ∈ ∆ .

74 Low hanging fruits

+ I Let ML (resp. ML ) be the multimodal logic into which ALC concepts (resp. knowledge bases) are translated.

I ML satisfiability problem is PSPACE-complete, it has the finite model property and the tree model property.

+ I ML satisfiability problem is EXPTIME-complete, it has the finite model property and the tree model property.

I ALC satisfiability problem is PSPACE-complete and it has the finite interpretation property.

I ALC knowledge base consistency problem is EXPTIME-complete and it has the finite tree-shaped interpretation property.

75 Recapitulation

76 Recapitulation: concept and role constructors

Name Syntax Semantics Top > ∆I Bottom ⊥ ∅ Conjunction C u D CI ∩ DI Disjunction C t D CI ∪ DI Negation ¬C ∆I \ CI Existential restr. ∃r.C {a ∈ ∆I | r I (a) ∩ CI 6= ∅} Value restr. ∀r.C {a ∈ ∆I | r I (a) ⊆ CI } Unqual. nb. restr. (≤ n r) {a ∈ ∆I | card({b | (a, b) ∈ r I } ≤ n)} Qual. nb. restr. (≤ n r · C) {a ∈ ∆I | card({b ∈ CI | (a, b) ∈ r I } ≤ n)} Nominal {a} {aI } Role value map r v s {a ∈ ∆I | r I (a) ⊆ sI (a)} Inverse role r − {(b, a) | (a, b) ∈ r I } Role composition r ◦ s {(a, b) | ∃ a0 (a, a0) ∈ r I and (a0, b) ∈ sI }

77 Recapitulation: Terminological and assertional axioms

Name Syntax Semantics General inclusion axiom C v D CI ⊆ DI Concept definition A ≡ C AI = CI Role inclusion r v s r I ⊆ sI Role transitivity Trans(r) r I is transitive Concept assertion a : C aI ∈ CI Role assertion (a, b): r (aI , bI ) ∈ r I

78 Conclusion

I Lecture 1 (today): Introduction to description logics

I Getting familiar with DL terminology.

I Playing with formulae and decision problems.

I Lecture 2: Tableaux proof systems and complexity .

I Complete calculi for ALC and variants.

I Complexity results.

I Undecidability.

I Lecture 3: Introduction to temporal logics for multi-agents systems.

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